The Role of Noise in Finite Ensembles of Nanomagnetic Particles
The Role of Noise in Finite Ensembles of Nanomagnetic Particles
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DOI：
10.1007/s00205-013-0654-4
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发表时间：
2013-07
影响因子：
2.5
通讯作者：
M. Neklyudov;A. Prohl
中科院分区：
文献类型：
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作者：
M. Neklyudov;A. Prohl
有限个纳米磁性粒子的动力学可用随机Landau-Lifshitz-Gilbert方程描述。我们证明了系统很快地以指数形式松弛到唯一的不变度量，它由Boltzmann分布来描述。我们提出了两种方法来验证这一结果。第一种是对马尔可夫链使用的一般理论(Meyn和Tweedy，Adv Appl Prob 24：542-574,1992；Meyn和Tweedy，Adv Appl Prob 25：487-517 1993；Meyn和Tweedy，Adv Appl Prob 25：518-548,1993)，这涉及到Lyapunov结构和转移概率的不可约性的概念；我们在上确界拓扑中证明了指数收敛，但缺乏显式的速度。第二种方法在弱L2拓扑下证明了指数遍历性，并且具有Arrhenius型定律的显式收敛速度。然后，我们讨论了两种隐式离散格式来逼近有限次和无限次的跃迁函数：第一种格式继承了单自旋的几何“单位长度”性质和Lyapunov结构，并证明了它是几何遍历的；此外，迭代以有限次的速度强收敛。第二种方案的计算效率更高，因为它是线性的；它被证明在所有有限次都以最优速度弱收敛。我们利用Shardlow和Stuart(Siam J Numer Anal 37(4)：1120-1137 2000)的一个一般结果，得出两个离散化的极限问题的不变测度的收敛。
The dynamics of finitely many nanomagnetic particles are described by the stochastic Landau–Lifshitz–Gilbert equation. We show that the system relaxes exponentially quickly to the unique invariant measure which is described by a Boltzmann distribution. We present two approaches to verify this result. The first uses the general theory of (Meyn and Tweedy, Adv Appl Prob 24:542–574, 1992; Meyn and Tweedy, Adv Appl Prob 25:487–517 1993; Meyn and Tweedy, Adv Appl Prob 25:518–548, 1993) for Markov chains, which involves the concepts of a Lyapunov structure, and irreducibility of transition probabilities; we show exponential convergence in a supremum topology, but lack explicit rates. The second approach shows exponential ergodicity in a weakerL2topology, with an explicit rate of convergence of the Arrhenius type law. Then, we discuss two implicit discretizations to approximate transition functions at both finite and infinite times: the first scheme is shown to inherit the geometric ‘unit-length’ property of single spins, as well as the Lyapunov structure, and is shown to be geometrically ergodic; moreover, iterates converge strongly with a rate for finite times. The second scheme is computationally more efficient since it is linear; it is shown to converge weakly at an optimal rate for all finite times. We use a general result of Shardlow and Stuart (Siam J Numer Anal 37(4):1120–1137 2000) to conclude convergence to the invariant measure of the limiting problem for both discretizations.